On Various R-duals and the Duality Principle

Diana T. Stoeva, Ole Christensen

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Abstract

The duality principle states that a Gabor system is a frame if and only if the corresponding adjoint Gabor system is a Riesz sequence. In general Hilbert spaces and without the assumption of any particular structure, Casazza, Kutyniok and Lammers have introduced the so-called R-duals that also lead to a characterization of frames in terms of associated Riesz sequences; however, it is still an open question whether this abstract theory is a generalization of the duality principle. In this paper we prove that a modified version of the R-duals leads to a generalization of the duality principle that keeps all the attractive properties of the R-duals. In order to provide extra insight into the relations between a given sequence and its R-duals, we characterize all the types of R-duals that are available in the literature for the special case where the underlying sequence is a Riesz basis.
Original languageEnglish
JournalIntegral Equations and Operator Theory
Volume84
Issue number4
Pages (from-to)577-590
ISSN0378-620X
DOIs
Publication statusPublished - 2016

Keywords

  • Duality principle
  • Frames
  • R-duals
  • R-duals of type II
  • R-duals of type III
  • Riesz bases
  • Riesz sequences

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